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Physics World

Measure for measure in quantum theory

F. E. Harrison

Published Mar 1, 1996• 2 citations
Publisher

Abstract

It is impossible to observe a quantum system, be it a particle or a light field, without disturbing it in some way. However, we can control to some extent how the system is disturbed by the measurement. In theory this allows us to measure certain observables without disturbing them, by shunting all the unavoidable back-action noise into some other observable. Not surprisingly, the observables form a complementary pair, obeying the uncertainty principle.

DOI
10.1088/2058-7058/9/3/22

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It is impossible to observe a quantum system, be it a particle or a light field, without disturbing it in some way. However, we can control to some extent how the system is disturbed by the measurement. In theory this allows us to measure certain observables without disturbing them, by shunting all the unavoidable back-action noise into some other observable. Not surprisingly, the observables form a complementary pair, obeying the uncertainty principle.
In quantum mechanics, measurements disturb the state of the system being measured. This disturbance is largest for complementary properties (e.g. position and momentum) and hence limits the precision with which such properties can be determined simultaneously. Often, this fact is conflated with Heisenberg's uncertainty principle, which refers to an uncertainty relation between complementary properties that is intrinsic to quantum states. In this thesis, the distinction between these two fundamental characteristics of quantum mechanics is made clear. At the intersection of the two are "joint measurements", which circumvent measurement disturbance to simultaneously determine complementary properties. They have applications in quantum metrology and enable a direct measurement of quantum states. The focus of this thesis is on the latter. The thesis is structured in the following way. The first chapter serves as an introduction to joint measurements. It surveys the seminal works in the field, doing so in a chronological manner to provide some historical context. The remainder of the thesis discusses two strategies to experimentally achieve joint measurements. The first strategy is to sequentially measure the complementary properties, making these measurements weak so that they do not disrupt each other. The second strategy is to first clone the system being measured, and then measure each complementary property on a separate clone. Both strategies are experimentally demonstrated on polarized photons, but can be readily extended to other systems.
One of the formulations of Heisenberg uncertainty principle, concerning so-called measurement uncertainty, states that the measurement of one observable modifies the statistics of the other. Here, we derive such a measurement uncertainty principle from two comprehensible assumptions: impossibility of instantaneous messaging at a distance (no-signaling), and violation of Bell inequalities (nonlocality). The uncertainty is established for a pair of observables of one of two spatially separated systems that exhibit nonlocal correlations. To this end, we introduce a gentle form of measurement which acquires partial information about one of the observables. We then bound disturbance of the remaining observables by the amount of information gained from the gentle measurement, minus a correction depending on the degree of nonlocality. The obtained quantitative expression resembles the quantum mechanical formulations, yet it is derived without the quantum formalism and complements the known qualitative effect of disturbance implied by nonlocality and no-signaling.
States and observables Bras, kets, and operators There are at least two fundamental assumptions made in classical physics which do not apply in quantum physics. These two assumptions concern the types of measurements we can perform on a given system. The first assumption made is that it is possible, at least in principle, to perform a measurement on a given system in a way in which the interaction between the system being studied and the measuring apparatus can be made as small as we like. Because the act of measurement will disturb the system in some way, it is assumed that we can compensate for this disturbance to minimize the interaction. For example, if we look visually at an object by illuminating it with light, the illumination caused by the light leaving the light source and striking the object before being reflected to our eyes will impart some momentum to the object. Given the wavelength of light used we can always (in classical physics) work out what momentum is imparted, and we can compensate for this change in momentum by applying an appropriate force to the object to counteract this change. The second assumption made in classical physics is that there are no restrictions on what measurements we can make on a system either in type or in accuracy. Both assumptions can break down when we consider systems which must be described by quantum mechanics.
The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of followed by a projective measurement of can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.